Timeline for Inference on the minimizing/maximizing value of $x$ in a regression
Current License: CC BY-SA 4.0
11 events
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| Jun 25, 2022 at 14:54 | comment | added | kjetil b halvorsen♦ | See stats.stackexchange.com/questions/161745/… for one approach! | |
| Dec 18, 2019 at 16:26 | answer | added | Andrew M | timeline score: 0 | |
| Dec 18, 2019 at 16:13 | comment | added | PauZen | One way to "measure" this rate in projection estimation (which is what you do here) is by having more or less parameter to estimate (so here you choose polynomials of order 2, could be more). It obviously depends of the number of elements in sample (here 30 if i understand) and also the class of function you want to estimate. To be honest, here with 30 samples there is no magic. Except if you know a lot about the function you want to estimate, you can't change the world. But i will get more information about argmin estimation, if i can find minimax rate for that. It's interesting. | |
| Dec 18, 2019 at 16:08 | comment | added | PauZen | I was editing my comment :D. Anyway you are right, if we change model this is tricky. I suppose this is because the estimation of argmin has need of estimating the function. I don't have proof for that but what you say seems true, even having the roots is quite useless since we don't know which one is the more little. Interesting question (just as a note, my point of view is the minimax optimality. Or if you prefer, what is the best rate you can have to estimate what you want in the worst situation under some assumption which are here the fact f is regular enough). | |
| Dec 18, 2019 at 16:06 | comment | added | Andrew M | @PauZen, I completely agree, and am uncomfortable about this. So my question is about model classes with weaker assumptions that would still allow testing for, and inference on a minimizing value of $f(x)$. | |
| Dec 18, 2019 at 16:03 | comment | added | PauZen | By your model (the quadratic fit) you assume by assumption a lot about the form of your function (especially about global minimum). | |
| Dec 17, 2019 at 20:49 | comment | added | Andrew M | @PauZen: Thanks for the comment. Under the quadratic model $E (f'(x)) = b + 2cx$ and indeed that is the quantity I am using for inference. However, I don't see how I can avoid an intersection test, and inference on a non-linear function of $(b, c)$, as I need to verify that both $\exists x \in \text{range}(X)_i: f'(x) = 0$ and $c>0$, since just finding a zero doesn't insure I found a local minimum, nor does the fact that $c>0$ mean that a minimum occurs. So an ideal model would provide inference on the the roots of $f'(x)$ for $f$ unimodal. | |
| Dec 17, 2019 at 20:38 | comment | added | Andrew M | @JamesPhillips In my view, the AUC34 = 0 are very important contributors, and eliminating them makes it much harder to estimate the functional relationship, since it dramatically reduces the variation in $x$. You might worry that there could a threshold effect, as do I. However, we address that in a separate analysis. | |
| Dec 17, 2019 at 13:29 | comment | added | James Phillips | From the plot, it appears that using values where AUC34 = 0 is not contributing to the overall model. If you separately model AUC34 equal to zero and AUC34 greater than zero, combining those separate models, would this be useful? | |
| Dec 17, 2019 at 9:50 | comment | added | PauZen |
I would estimate the first derivative of f(assuming its regular enough). And you can use any base to do your estimation (here you use polynomial, it could be any functional base, as long as you have few parameter to estimate this is not a problem). I don't have time to dig further, but i can say one last important thing. Its always better to directly estimate what we want to estimate and not do multiple estimation step, its less precise and hide many problems.
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| Dec 17, 2019 at 6:43 | history | asked | Andrew M | CC BY-SA 4.0 |